Correlation Functions of CMB Anisotropy and Polarization Kin-Wang Ng∗ and Guo-Chin Liu†
arXiv:astro-ph/9710012v2 25 Feb 1998
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C. We give a full analysis of the auto- and cross-correlations between the Stokes parameters of the cosmic microwave background. In particular, we derive the windowing function for an antenna with Gaussian response in polarization experiment, and construct correlation function estimators corrected for instrumental noise. They are applied to calculate the signal to noise ratios for future anisotropy and polarization measurements. While the small-angular-scale anisotropy-polarization correlation would be likely detected by the MAP satellite, the detection of electric and magnetic polarization would require higher experimental sensitivity. For large-angular-scale measurements such as the being planned SPOrt/ISS, the expected signal to noise ratio for polarization is greater than one only for reionized models with high reionization redshifts, and the ratio is less for anisotropypolarization correlation. Correlation and covariance matrices for likelihood analyses of ground-based and satellite data are also given. PACS numbers: 98.70.Vc, 98.80.Es
I. INTRODUCTION
The detection of the large-angle anisotropy of the cosmic microwave background (CMB) by the COBE DMR experiment [1] provided important evidence of large-scale spacetime inhomogenities. Since then, a dozen of smallscale anisotropy measurements have hinted that the Doppler peak resulting from acoustic oscillations of the baryonphoton plasma on the last scattering surface seems to be present [2]. CMB measurements gain an advantage over other traditional observations due to the fact that the small CMB fluctuations can be well treated as linear, while the low-redshift universe is in a non-linear regime. It is now well established that CMB temperature anisotropies are genuine imprint of the early universe, which could potentially be used to determine to a high precision virtually all cosmological parameters of interest. It has been estimated that a number of cosmological parameters can be determined with standard errors of 10% or better by the upcoming NASA MAP satellite [3]. Furthermore, the future Planck Surveyor CMB mission would have capability of observing the early universe about 100 times better than MAP. At this point, we should explore as much information as possible besides the temperature anisotropy contained in the relic photons. Anisotropic radiation possessing a non-zero quadrupole moment acquires a net linear polarization when it is scattered with electrons via Thomson scattering [4] (also see Eq. (6) of Ref. [5]). When the photons begin to decouple from the matter on the last scattering surface and develope a quadrupole anisotropy via Sachs-Wolfe effect [6], linear polarization is created from scatterings with free electrons near the last scattering surface. Studies have shown that on small angular scales the rms polarization, in a standard universe, is a few percents of the rms anisotropy, while the large-scale polarization is insignificant [7]. In models with early reionization, the large-scale polarization is greatly enhanced, to a few percents level, but the small-scale anisotropy is suppressed significantly [5,8]. Therefore, CMB polarization would provide a valuable complementary information to the anisotropy measurements. In addition, the anisotropy-polarization cross correlation offers a test of physics on the last scattering surface, as well as a possibility of distinguishing the scalar and tensor perturbations [9,10]. However, all of these polarization calculations have relied on a small-angle approximation, which may not be valid when a large sky-coverage is considered. As such, full-sky analyses of the polarization have been performed [11–15]. It was found that there are modifications to low multipole moments (l < 30) of the polarization power spectra, where the tensor contribution dominates over the scalar contribution [11,14]. More importantly, rotationally invariant power spectra of the Stokes parameters have been constructed [12–15]. In particular, one of them is a parity-odd magnetic polarization spectrum, which vanishes for scalar-induced polarization, thereby allowing one to make a model-independent identification of non-scalar (i.e. vector or tensor) perturbations (also see Ref. [16]). Recently it was shown that magnetic polarization would be a strong discriminator between defect and inflation models [17,18]. Also, a new physically transparent formalism based
∗ †
[email protected] [email protected]
1
on the total angular momentum representation [19] was proposed [17,20], which simplifies the radiative transport problem and can be easily generalized to open universes [21]. Since polarization fluctuations are typically at a part in a million, an order magnitude below the temperature fluctuations, to measure this signal requires high detector sensitivity, long integration time, and/or a large number of pixels. So far, only experimental upper limits have been obtained [22–24], with the current limit on the linear polarization being 16µK [24]. Ground-based experiments being planned or built will probably achieve detection sensitivity using low-noise HEMT amplifiers as well as long hours of integration time per pixel. The MAP satellite will launch in 2000 and make polarization measurements of the whole sky in about 105 pixels. If the polarization foreground can be successfully removed, MAP should marginally reach the detection level. For a detection of the magnetic polarization one would require either several years of MAP observations or the Planck mission [12,18,25]. We expect that polarization measurements are as important as anisotropy in future missions. Previous full-sky studies of the polarization are mainly based on angular power spectrum estimators in Fourier space. Although electric- and magnetic-type scalar fields E and B in real space can be constructed, they must involve nonlocal derivatives of the Stokes components. In this paper, we will study in detail the auto- and crosscorrelation functions of the Stokes parameters themselves in real space. Although the two approaches should be equivalent to each other, one can find individual advantages in different situations. We will follow the formalism of Ref. [12], expanding the Stokes parameters in terms of spin-weighted spherical harmonics. The expansion coefficients are rotationally invariant power spectra which will be evaluated using CMBFAST Boltzmann code developed by Seljak and Zaldarriaga [26]. In Sec. II we briefly introduce the CMB Stokes parameters and their relation to spinweighted spherical harmonics. Sec. III is devoted to discussions of the properties of the harmonics, the harmonics representation of rotation group, and the generalized addition theorem and recursion relation. In Sec. IV we expand the Stokes parameters in spin-weighted harmonics, and briefly explain how to compute the power spectra induced by scalar and tensor perturbations. In Sec. V we derive window functions appropriate to detectors with Gaussian angular response in anisotropy and polarization experiments. The instrumental noise of detectors in CMB measurements is treated in Sec. VI as white noise superposed upon the microwave sky. Sec. VII is to construct the auto- and crosscorrelation function estimators corrected for noise bias in terms of the power spectra. As examples, in Sec. VIII we compute the means and variances of the estimators for different configurations of future space missions in standard cold dark matter models. Further, we outline the likelihood analysis of the experimental data in Sec. IX. Sec. X is our conclusions. II. STOKES PARAMETERS
Polarized light is conventionally described in terms of the four Stokes parameters (I, Q, U, V ), where I is the intensity, Q and U represent the linear polarization, and V describes the circular polarization. Each parameter is a function of the photon propogation direction n ˆ . Let us define T = I − I¯
(2.1)
as the temperature fluctuation about the mean. Since circular polarization cannot be generated by Thomson scattering alone, V decouples from the other components. So, it suffices to consider only the Stokes components (T, Q, U ) as far as CMB anisotropy and polarization is concerned. Traditionally, for radiation propagating radially along eˆr in the spherical coordinate system, see Fig. 1, Q and U are defined with respect to an orthonormal basis (ˆ a, ˆb) on the sphere, which are related to (ˆ eθ , eˆφ ) by a ˆ = eˆφ ,
and ˆb = −ˆ eθ .
(2.2) √ ˆ Then, Q is the ˆ directions, while U is the difference in the (ˆ a + ˆb)/ 2 √ difference in intensity polarized in the b and a and (ˆ a − ˆb)/ 2 directions [27]. Under a left-handed rotation of the basis about eˆr through an angle ψ, ′ a ˆ a ˆ cos ψ − sin ψ (2.3) ˆb , ˆb′ = sin ψ cos ψ or equivalently, 1 1 ′ √ a ˆ + iˆb′ = eiψ √ a ˆ + iˆb . 2 2 Under this transformation T and V are invariant while Q and U being transformed to [27] 2
(2.4)
Q′ U′
=
cos 2ψ sin 2ψ − sin 2ψ cos 2ψ
Q U
,
(2.5)
which in complex form is Q′ (ˆ er ) ± iU ′ (ˆ er ) = e∓2iψ [Q(ˆ er ) ± iU (ˆ er )] .
(2.6)
Hence, Q(ˆ n) ± iU (ˆ n) has spin-weight ∓2.1 Therefore, we may expand each Stokes parameter in its appropriate spin-weighted spherical harmonics [19,12]. Unfortunately, the convention in theory has a little difference from experimental practice. In CMB polarization measurements, usually the north celestial pole is chosen as the reference axis eˆ3 , and linear polarization at a point x ˆ on the celestial sphere is defined by Q(ˆ x) = TN,S − TE,W ,
and U(ˆ x) = TN E,SW − TN W,SE ,
(2.7)
where TN,S is the antenna temperature of radiation polarized along the north-south direction, and so on [22]. In small-scale experiments covering only small patches of the sky, the geometry is essentially flat, so one can simply choose any local rectangular coordinates to define Q and U. Since an observation in direction x ˆ receives radiation with propagating direction n ˆ = −ˆ x, we have Q(ˆ x) = Q(ˆ n),
and U(ˆ x) = −U (ˆ n).
(2.8)
III. SPIN-WEIGHTED SPHERICAL HARMONICS
An explicit expression of spin-s spherical harmonics is
2
[28,29]
1 θ 2l + 1 (l + m)! (l − m)! 2 2l sin s Ylm (θ, φ)= (−1) e 4π (l + s)! (l − s)! 2 X l − s l + s θ l−s−r 2r+s−m (−1) cot × , r+s−m r 2 m imφ
(3.1)
r
where max(0, m − s) ≤ r ≤ min(l − s, l + m).
(3.2)
Note that the common spherical harmonics Ylm = 0 Ylm . They have the conjugation relation and parity relation: ∗ s Ylm (θ, φ)
s Ylm (π
= (−1)m+s −s Yl−m (θ, φ),
− θ, φ + π) = (−1)l −s Ylm (θ, φ).
They satisfy the orthonormality condition and completeness relation: Z dΩ s Yl∗′ m′ (θ, φ) s Ylm (θ, φ) = δl′ l δm′ m , X lm
∗ ′ ′ s Ylm (θ , φ ) s Ylm (θ, φ)
= δ(φ′ − φ)δ(cos θ′ − cos θ).
Therefore, a quantity η of spin-weight s defined on the sphere can be expanded in spin-s basis,
1 2
Generally, a quantity η will be said to have spin-weight s if it transforms as η ′ = esiψ η under the rotation (2.4) [28]. In Ref. [28], the sign (−1)m is absent. We have added the sign in order to match the conventional definition for Ylm .
3
(3.3) (3.4)
(3.5)
(3.6)
η(θ, φ) =
X
ηlm s Ylm (θ, φ),
(3.7)
lm
where the expansion coefficients ηlm are scalars. ¯′ , acting on η of spin-weight s, are defined by [28] The raising and lowering operators, ∂′ and ∂ ∂ ∂ ′ s (sin θ)−s η, + i csc θ ∂η = −(sin θ) ∂θ ∂φ ∂ ∂ −s ′ ¯ ∂η = −(sin θ) (sin θ)s η. − i csc θ ∂θ ∂φ
(3.8) (3.9)
When they act on the spin-s spherical harmonics, we have [28] 1
∂′ s Ylm = [(l − s)(l + s + 1)] 2 s+1 Ylm , 1 ¯′ s Ylm = − [(l + s)(l − s + 1)] 2 s−1 Ylm , ∂ ¯′ ∂′ s Ylm = −(l − s)(l + s + 1) s Ylm . ∂
(3.10) (3.11) (3.12)
Using these raising and lowering operations, we obtain the generalized recursion relation for l − 2 ≥ max(|s|, |m|),
l+s l−s
12
s Ylm
(2l + 1)(2l − 1) = (l + m)(l − m)
21
cos θ s Yl−1,m
1 (2l + 1)(l + m − 1)(l − m − 1)(l − s − 1) 2 − s Yl−2,m (2l − 3)(l + m)(l − m)(l + s − 1) 21 (2l + 1)(2l − 1) +s sin θ s−1 Yl−1,m . (l + m)(l − m)(l − s)(l + s − 1)
(3.13)
This will be used for evaluating the correlation functions in Sec. VIII. Table 1 lists explicit expressions for some low-l spin-weighted harmonics, from which higher-l ones can be constructed. The harmonics are related to the representation matrices of the 3-dimensional rotation group. If we define a rotation R(α, β, γ) as being composed of a rotation α around eˆ3 , followed by β around the new eˆ′2 and finally γ around eˆ′′3 , the rotation matrix of R will be given by [28] r 4π −isγ l . (3.14) D−sm (α, β, γ) = s Ylm (β, α)e 2l + 1 Let us consider a rotation group multiplication, R(α, β, −γ) = R(φ′ , θ′ , 0)R−1 (φ, θ, 0),
(3.15)
where the angles are defined in Fig. 1. In terms of rotation matrices, it becomes X Dsl 1 s2 (α, β, −γ) = Dsl 1 m (φ′ , θ′ , 0)Dsl∗2 m (φ, θ, 0),
(3.16)
m
which leads to the generalized addition theorem,3 X m
∗ ′ ′ s1 Ylm (θ , φ ) s2 Ylm (θ, φ)
=
r
2l + 1 (−1)s1 −s2 4π
3
−s1 Yls2 (β, α)e
−is1 γ
.
(3.17)
This theorem was first derived in Eq. (7) of Ref. [17], which however does not give correct signs for the geometric phase angles, α and γ. Eq. (3.17) will be useful in the following sections.
4
IV. POWER SPECTRA
Following the notations in Ref. [12], we expand the Stokes parameters as X T (ˆ n) = aT,lm Ylm (ˆ n), lm
X
Q(ˆ n) − iU (ˆ n) =
a2,lm 2 Ylm (ˆ n),
lm
X
Q(ˆ n) + iU (ˆ n) =
a−2,lm −2 Ylm (ˆ n).
(4.1)
a∗−2,lm = (−1)m a2,l−m .
(4.2)
lm
The conjugation relation (3.3) requires that a∗T,lm = (−1)m aT,l−m ,
For Stokes parameters in CMB measurements, using the parity relation (3.4), we have X T (ˆ x) = (−1)l aT,lm Ylm (ˆ x), lm
Q(ˆ x) + iU(ˆ x) =
Q(ˆ x) − iU(ˆ x) =
X
(−1)l a2,lm −2 Ylm (ˆ x),
lm
X
(−1)l a−2,lm 2 Ylm (ˆ x).
(4.3)
lm
Isotropy in the mean guarantees the following ensemble averages:
∗ aT,l′ m′ aT,lm = CT l δl′ l δm′ m ,
∗ a2,l′ m′ a2,lm = (CEl + CBl )δl′ l δm′ m ,
∗ a2,l′ m′ a−2,lm = (CEl − CBl )δl′ l δm′ m ,
∗ aT,l′ m′ a2,lm = −CCl δl′ l δm′ m .
(4.4)
Consider two points n ˆ ′ (θ′ , φ′ ) and n ˆ (θ, φ) on the sphere. Using the addition theorem (3.17) and Eq. (4.4), we obtain the correlation functions, hT ∗ (ˆ n′ )T (ˆ n)i =
X 2l + 1 l
4π
CT l Pl (cos β),
(4.5)
s
(l − 2)! CCl Pl2 (cos β)e2iα , (l + 2)! l r X 2l + 1 ′ ′ ∗ h[Q(ˆ n ) + iU (ˆ n )] [Q(ˆ n) + iU (ˆ n)]i = (CEl + CBl ) 2 Yl−2 (β, 0)e2i(α−γ) , 4π l r X 2l + 1 ′ ′ ∗ (CEl − CBl ) 2 Yl2 (β, 0)e2i(α+γ) , h[Q(ˆ n ) − iU (ˆ n )] [Q(ˆ n) + iU (ˆ n)]i = 4π
hT ∗ (ˆ n′ )[Q(ˆ n) + iU (ˆ n)]i = −
X 2l + 1 4π
(4.6) (4.7) (4.8)
l
where α, β, and γ are the angles defined in Fig. 1. Eq. (4.6) is the most general form of those found in Refs. [9,11,16,30]. In the small-angle approximation, i.e. β
